Lemma REF  leads to \(\Omega (n^{-1}\Vert x\Vert _2)\le \Omega (n^{-1}\Vert x\Vert _1) = \Omega \left(n^{-1}\sum _{i=1}^{n}x_i\right)\) . Moreover, for i) \(\Omega \left(n^{-1}\sum _{i=1}^{n}x_i\right)\le \sum _{i=1}^n\Omega (n^{-1}x_i)\le n^{-d}\sum _{i=1}^n\Omega (x_i)\) . For ii), \(\Omega \left(n^{-1}\sum _{i=1}^{n}x_i\right)\le n^{-1}\sum _{i=1}^n\Omega (x_i)\)  due to Jensen's inequality of convex functions [1]}. Hence, \(\Omega (z) = \hat{\Omega }(z)\)  complies with (REF ) for either i) or ii).
